Write down Stokes' theorem for a vector field $\mathbf{A}(\mathbf{x})$ on $\mathbb{R}^{3}$.

Let the surface $S$ be the part of the inverted paraboloid

$$z=5-x^{2}-y^{2}, \quad 1<z<4$$

and the vector field $\mathbf{A}(\mathbf{x})=\left(3 y,-x z, y z^{2}\right)$.

(a) Sketch the surface $S$ and directly calculate $I=\int_{S}(\nabla \times \mathbf{A}) \cdot d \mathbf{S}$.

(b) Now calculate $I$ a different way by using Stokes' theorem.